A Matrix Exponential Generalization of the Laplace Transform of Poisson Shot Noise

We consider a generalization of the Laplace transform of Poisson shot noise defined as an integral transform with respect to a matrix exponential. We denote this integral transform as the matrix Laplace transform given its similarity to the Laplace-Stieltjes transform. We establish that the matrix Laplace transform is in general a natural matrix function extension of the typical scalar Laplace transform, and that the matrix Laplace transform of Poisson shot noise admits an expression that is analogous to the expression implied by Campbell's theorem for the Laplace functional of a Poisson point process. We demonstrate the utility of our generalization of Campbell's theorem in two important applications: the characterization of a Poisson shot noise process and the derivation of the complementary cumulative distribution function (CCDF) of signal to interference and noise (SINR) models with phase-type distributed fading powers. In the former application, we demonstrate how the higher order moments of a linear combination of samples of a Poisson shot noise process may be obtained directly from the elements of its matrix Laplace transform. We further show how arbitrarily tight approximations and bounds on the CCDF of this object may be obtained from the summation of the first row of its matrix Laplace transform. For the latter application, we show how the CCDF of SINR models with phase-type distributed fading powers may be obtained in terms of an expectation of the matrix Laplace transform of the interference and noise, analogous to the canonical case of SINR models with Rayleigh fading.